A quick way to convert between miles and kilometers is by using the fibonacci sequence. 1 1 2 3 5 8 13 21 34 55 89 etc. For miles, use the number to the right. 5 becomes 8. 500 becomes 800 (the actual number is 804.67... pretty darn close).For km to miles, use the number to the left. So 3400 kilometers is about 2100 miles. It's actually 2112.66 miles but that's only a difference of < 1%.If you have a number that doesn't quite fit in the sequence, such as 100 km, just break it down into numbers that do fit. 100 = 89 + 8 + 3. So that becomes 55 + 5 + 2 = 62 miles. 100km is actually 62.1371 miles. It's quite a nice trick to know.

In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:[1][2]

(sequence A000045 in OEIS)By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation with seed values[3]

The Fibonacci sequence is named after Leonardo Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics,[4] although the sequence had been described earlier in Indian mathematics.[5][6][7] By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber Abaci began the sequence with F1 = 1, without an initial 0.

Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings,[8] such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple,[9] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.[10]

Origins[edit]

A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Latin and Roman numerals and the value in Hindu-Arabic numerals.The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[6][11] In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonacci number Fm + 1.[7]

Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150)".[5] Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites scholars who interpret it in context as saying that the cases for m beats (Fm+1) is obtained by adding a [S] to Fm cases and [L] to the Fm−1 cases. He dates Pingala before 450 BC.[12]

However, the clearest exposition of the series arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].[13]The series is also discussed by Gopala (before 1135 AD) and by the Jain scholar Hemachandra (c. 1150).

In the West, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci.[4] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

At the end of the first month, they mate, but there is still only 1 pair.At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.[14]

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[15]

Applications[edit]

The Fibonacci numbers are important in the computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.[46]

Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.

The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. Specifically, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.

Fibonacci numbers are used by some pseudorandom number generators.

Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.

Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.

The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.

The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as µ-law.[48][49]

Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.[50]

In nature[edit]

Further information: Patterns in nature and Phyllotaxis

Yellow Chamomile head showing the arrangement in 21 (blue) and 13 (aqua) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.Fibonacci sequences appear in biological settings,[8] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[9] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.[10] In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g., relating to the breeding of rabbits in Fibonacci's own unrealistic example, the seeds on a sunflower, the spirals of shells, and the curve of waves.[51] The Fibonacci numbers are also found in the family tree of honeybees.[52]

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.[53]

There is a large amount of information here on the Fibonacci Numbers and related series and the on the Golden section, so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature.

Fibonacci Numbers and Golden sections in Nature

Ron Knott was on Melvyn Bragg's In Our Time on BBC Radio 4, November 29, 2007 when we discussed The Fibonacci Numbers (45 minutes). You can listen again online or download the podcast. It is a useful general introduction to the Fibonacci Numbers and the Golden Section.Fibonacci Numbers and Nature Fibonacci and the original problem about rabbits where the series first appears, the family trees of cows and bees, the golden ratio and the Fibonacci series, the Fibonacci Spiral and sea shell shapes, branching plants, flower petal and seeds, leaves and petal arrangements, on pineapples and in apples, pine cones and leaf arrangements. All involve the Fibonacci numbers - and here's how and why. The Golden section in Nature Continuing the theme of the first page but with specific reference to why the golden section appears in nature. Now with a Geometer's Sketchpad dynamic demonstration. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

The Puzzling World of Fibonacci Numbers

A pair of pages with plenty of playful problems to perplex the professional and the part-time puzzler!The Easier Fibonacci Puzzles page has the Fibonacci numbers in brick wall patterns, Fibonacci bee lines, seating people in a row and the Fibonacci numbers again, giving change and a game with match sticks and even with electrical resistance and lots more puzzles all involve the Fibonacci numbers!The Harder Fibonacci Puzzles page still has problems where the Fibonacci numbers are the answers - well, all but ONE, but WHICH one? If you know the Fibonacci Jigsaw puzzle where rearranging the 4 wedge-shaped pieces makes an additional square appear, did you know the same puzzle can be rearranged to make a different shape where a square now disappears?For these puzzles, I do not know of any simple explanations of why the Fibonacci numbers occur - and that's the real puzzle - can you supply a simple reason why??The Intriguing Mathematical World of Fibonacci and Phi

The golden section numbers are also written using the Greek letters Phi and phi .The Mathematical Magic of the Fibonacci numbers looks at the patterns in the Fibonacci numbers themselves: the Fibonacci numbers in Pascal's Triangle; using the Fibonacci series to generate all right-angled triangles with integers sides based on Pythagoras Theorem.An auxiliary page:More on Pythagorean trianglesIf you want to look like a number wizard to your friends then try out the simple Fibonacci numbers trick!The following pages give you lots of opportunities to find your own patterns in the Fibonacci numbers. We start with a complete list of...The first 500 Fibonacci numbers... completely factorized up to Fib(300) and all the prime Fibonacci numbers are identified up to Fib(500).A Formula for the Fibonacci numbers Is there a direct formula to compute Fib(n) just from n? Yes there is! This page shows several and why they involve Phi and phi - the golden section numbers.Fibonacci bases and other ways of representing integers We use base 10 (decimal) for written numbers but computers use base 2 (binary). What happens if we use the Fibonacci numbers as the column headers?The Golden Section

The golden section number is closely connected with the Fibonacci series and has a value of (5 + 1)/2 or:1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

which we call Phi (note the capital P) on these pages. The other number also called the golden section is Phi-1 or 0·61803... with exactly the same decimal fraction part as Phi. This value we call phi (with a small p) here. Phi and phi have some interesting and unique properties such as 1/phi is the same as 1+phi=Phi. The third of Simon Singh's Five Numbers programmes broadcast on 13 March 2002 on BBC Radio 4 was all about the Golden Ratio. It is an excellent introduction to the golden section. I spoke on it about the occurrence in nature of the golden section and also the Change Puzzle. Hear the whole programme (14 minutes) using the free RealOne Player.The Golden section and Geometry The golden section is also called the golden ratio, the golden mean and the divine proportion.

Two more pages look at its applications in Geometry: first in flat (or two dimensional) geometry and then in the solid geometry of three dimensions.Two-dimensional Geometry and the Golden section or Fantastic Flat Facts about Phi See some of the unexpected places that the golden section (Phi) occurs in Geometry and in Trigonometry: pentagons and decagons, paper folding and Penrose Tilings where we phind phi phrequently!An auxiliary page on Exact Trig Values for Simple Angles explores the many places that Phi and phi occur when we try to find the exact values of the sines, cosines and tangents of simple angles like 36° and 54°.The Golden Geometry of Solids or Phi in 3 dimensions The golden section occurs in the most symmetrical of all the three-dimensional solids - the Platonic solids. What are the best shapes for fair dice? Why are there only 5?

The next pages are about the numbers Phi = 1·61803.. and phi = 0·61803... and their properties.Phi's Fascinating Figures - the Golden Section number All the powers of Phi are just whole multiples of itself plus another whole number. Did you guess that these multiples and the whole numbers are, of course, the Fibonacci numbers again? Each power of Phi is the sum of the previous two - just like the Fibonacci numbers too.Introduction to Continued Fractions is an optional page that expands on the idea of a continued fraction (CF) introduced in the Phi's Fascinating Figures page.There is also a Continued Fractions Converter (a web page - needs no downloads or special plug-is) to change decimal values, fractions and square-roots into and from CFs.This page links to another auxiliary page on Simple Exact Trig values such as cos(60°)=1/2 and finds all simple angles with an exact trig expression, many of which involve Phi and phi.Phigits and Base Phi Representations We have seen that using a base of the Fibonacci Numbers we can represent all integers in a binary-like way. Here we show there is an interesting way of representing all integers in a binary-like fashion but using only powers of Phi instead of powers of 2 (binary) or 10 (decimal).The Golden String

The golden string is also called the Infinite Fibonacci Word or the Fibonacci Rabbit sequence. There is another way to look at Fibonacci's Rabbits problem that gives an infinitely long sequence of 1s and 0s called the Golden String:-1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...

This string is a closely related to the golden section and the Fibonacci numbers.Fibonacci Rabbit Sequence See show how the golden string arises directly from the Rabbit problem and also is used by computers when they compute the Fibonacci numbers. You can hear the Golden sequence as a sound track too.The Fibonacci Rabbit sequence is an example of a fractal - a mathematical object that contains the whole of itself within itself infinitely many times over.Fibonacci - the Man and His Times

Who was Fibonacci? Here is a brief biography of Fibonacci and his historical achievements in mathematics, and how he helped Europe replace the Roman numeral system with the "algorithms" that we use today.Also there is a guide to some memorials to Fibonacci to see in Pisa, Italy.More Applications of Fibonacci Numbers and Phi

The Fibonacci numbers in a formula for Pi () There are several ways to compute pi (3·14159 26535 ..) accurately. One that has been used a lot is based on a nice formula for calculating which angle has a given tangent, discovered by James Gregory. His formula together with the Fibonacci numbers can be used to compute pi. This page introduces you to all these concepts from scratch.Fibonacci Forgeries Sometimes we find series that for quite a few terms look exactly like the Fibonacci numbers, but, when we look a bit more closely, they aren't - they are Fibonacci Forgeries.Since we would not be telling the truth if we said they were the Fibonacci numbers, perhaps we should call them Fibonacci Fibs !!The Lucas Numbers Here is a series that is very similar to the Fibonacci series, the Lucas series, but it starts with 2 and 1 instead of Fibonacci's 0 and 1. It sometimes pops up in the pages above so here we investigate it some more and discover its properties.It ends with a number trick which you can use "to impress your friends with your amazing calculating abilities" as the adverts say. It uses facts about the golden section and its relationship with the Fibonacci and Lucas numbers.The first 200 Lucas numbers and their factors together with some suggestions for investigations you can do.2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ....

The Fibonomials The basic relationship defining the Fibonacci numbers is F(n) = F(n – 1) + F(n – 2) where we use some combination of the previous numbers (here, the previous two) to find the next. Is there such a relationship between the squares of the Fibonacci numbers F(n)2? or the cubes F(n)3? or other powers? Yes there is and it involves a triangular table of numbers with similar properties to Pascal's Triangle and the binomial numbers: the Fibonomials.General Fibonacci Series The Lucas numbers change the two starting values of the Fibonacci series from 0 and 1 to 2 and 1. What if we changed these to any two values? These General Fibonacci series are called the G series but the Fibonacci series and Phi again play a prominent role in their mathematical properties. Also we look at two special arrays (tables) of numbers, the Wythoff array and the Stolarsky array and show how a these two collections of general Fibonacci series contain each whole number exactly once. The secret behind such clever arrays is ... the golden section number Phi!Fibonacci and Phi in the Arts

Fibonacci Numbers and The Golden Section In Art, Architecture and Music The golden section has been used in many designs, from the ancient Parthenon in Athens (400BC) to Stradivari's violins. It was known to artists such as Leonardo da Vinci and musicians and composers, notably Bartók and Debussy. This is a different kind of page to those above, being concerned with speculations about where Fibonacci numbers and the golden section both do and do not occur in art, architecture and music. All the other pages are factual and verifiable - the material here is a often a matter of opinion. What do you think?Reference

Fibonacci and Phi Formulae A reference page of about 300 formulae and equations showing the properties of the Fibonacci and Lucas series, the general Fibonacci G series and Phi. Also available in PDF format (21 pages) for which you will need the free Acrobat PDF Reader or plug-in for your browser.Links and Bibliography Links to other sites on Fibonacci numbers and the Golden section together with references to books and articles.